How can one tell when the definitions are in conjunction with the hypotheses?

The definitions have to be very well stated so there is no fuzzyness in what they connote. This is called well-defined. One of the best ways to get well-defined objects in mathematics is with axioms. You set up a small number of statements, called axioms, describing the thing you want to prove theorems about, and then you prove two special theorems:
1. The axioms don't contradict each other. (Consistency)
2. There actually is something that the axioms describe (Existence)

If you can do that, the object is well-defined, and you can make theorems about it by referrring to the axioms. By view of your consistency proof, this won't lead you to contradictions, and by view of the existence proof, you won't be talking about nothing at all.

If I were to use Newton's work on Principa as a example. He used 8 definitions before he wrote his 3 axioms or laws of motion. Now I have written an equation on a natural phenomenon and I am wondering what parts I should define before I start writing the axioms. Actually I already started writing the axioms but I feel like I am ahead of myself. Because I think the definition will give the proof a clear understanding of the natural phenomenon.